It has been difficult to parse your posts. English seems not to be your native language.
Some of the subsequent posts have made some things more clear. For instance, in english, an artificial intelligence is known as an AI, not an IA.

We have 3 ontological options:
1) We do not live now in a computational Simulation, and in the future it will not happen.
... — Belter

You need to make clear what you mean by 'live in' or by a simulation.
One case is a virtual reality (VR) where the experience is simulated to a real experiencer. This is already done today in any first-person video game. While playing the game, we 'live in' a computational simulation. A super AI is not required for this. My computer plays minecraft, a reasonable example of a first-person VR. While playing it, I remember that I am actually human. I have memory of being in a different world than the minecraft one.

The second case is a true simulation where a world is simulated down to the granularity required. If we are simulatable in that way, it would mean that all of physics (most in particular ourselves) can be mathematically described. It means no dualistic minds, else the simulated humans would lack them and not act at all like humans in the same way that a simulated radio (such simulations exist) will not be able to actually pick up real radio stations.

I've read all your posts, and it is not really clear which scenario.
In the VR case, why do I not have memory of being outside it? If always in it, how was I born?

In the pure-simulation case, our physics cannot be simulated by our physics, as pointed out by apokrisis. It would have to be run in a world with completely different physics which would permit a simulation of significantly higher cardinality of capacity, but the speed of it running would be irrelevant.
Michael Ossipoff's comments are about only the pure simulation case, and they are relevant. A simulation of world X allows X to be viewed by the runners of the simulation, but it doesn't really create X. It isn't ontology that is being done. So while I am capable of being simulated in some more capable universe, I am nevertheless not a simulation

[/quote]If the Simulation is made by a IA higher than human, and we define it as the ability of lying a normal human (following the idea of the Turing test), [/quote]First of all, a post-singularity AI might pass a Turing test, be we will very much know it is there. The singularity is by definition the point at which it can make self-improvements at a pace greater than improvements made by humanity. It has nothing to do with a Turing test, which is just a test to mimic a human in a text exchange, like this forum.

then the three options have the same probability a priori of be true, due to by definition a higher intelligence system can occult its existence.

That is not true by definition, and even if it was, it does not make even the odds of your three options. In fact, the physics we know are incapable of creation of such a self-simulation, so if we are in some sort of VR, it is being run in a universe of more computationally capable physics.

My claim is that Singularity is by definition unknown, such as God in religions, the player in video games, etc. A super IA is by definition, able to occult its existence. — Belter

The definition says nothing of the sort. The definition is that it no longer requires humans for improvement.

It turns out that an AI is not needed at all. The universe is purely mathematical in nature and can be simulated with a very simple machine with large enough capacity to hold the relevant state. The capacity is why it cannot be done with our own physics. But simple mathematics does not require an AI. The Babbage engine was one of the first to perform arithmetic faster than the rooms full of people with paper and slide rules, and it was hardly an AI.

In a pure simulation (not in the VR case), yes, the device doing the simulation would be occulted from the things being simulated, else it would be a simulation of a different thing.

Oh, and since you know who Max Tegmark is, perhaps this video will help a little. Pay particular attention to his remarks about how time probably doesn't move from one planck time to the next. If all of time is contained within a singularity of time, and the flow of time is just an illusion, there is no 'one planck time to the next' necessary. — StuartL

Tegmark does say that time probably does not flow, but he described time as a 4th dimension in addition to the 3 of space (not the same way you are using 'dimension'), and not as being contained in a singularity any more than is space is thus contained. The big-bang singularity is just one point (event) in spacetime.

I suspect that you don't know what is being described by time not flowing, and perhaps this is the source of confusion in your posts. It is not a suggestion that time is 'stuck' at one point (the singularity say) and the rest being just potential. No, all of spacetime (all points in space and in our past and future) share equal ontology, and there is no particular moment that is the present one. Flowing time is the assertion of that special addition to the model, the present. It supposedly moves (or is stuck, the way you seem to envision it). But the block view says there is no present at all. There is no one state that is the actual current one. All states are equally real.

This was all suggested about a century ago by Minkowski who originated the idea of spacetime, or at least formalized a lot of the mathematics behind it.

Commenting first on the most recent post, since some parts of the picture are being clarified.
I see something like an argument that there is an unfertilized egg and all these sperm could come along representing potential people. If one wins, it renders nonexistent all the other potential people. But it doesn't render the whole pregnant woman nonexistent. That part is confusing.

A singularity of pure energy exists in a its own time dimension (imaginary time).

Nothing else exists within imaginary time. — StuartL

How is there this imaginary time if there is no change? You have a sole existent with zero difference from one moment to the next, which is indistinguishable from no time at all.
You have a strange mixture of flowing and block time, and of existence and nonexistence. Why does this imaginary time seem to flow, but our time does not? Our spacetime doesn't exist, except as potential of an actual singularity.

The singularity is spherical.

Then it isn't a singularity, which would be a dimensionless point. A sphere has a spatial radius.

Therefore; the singularity possesses potential future properties for each possible interaction upon its surface. Given that the properties of the surface of a sphere are infinite, the possible resultant potential futures for the singularity are also infinite.

By infinite properties, you mean there are infinite way in which it could be struck, which would be true of any object, sphere or not. It would require a second existent with which to strike it, but you say that doesn't exist.

A multiverse of potential futures for the singularity exists, but should any of them get “banged” into actual events, the singularity and its dimension of time would cease to exist.

Why do they cease to exist? Why propose that?

If an infinite number of universes are possible, how come this one with us in it exists, if God didn't make it specifically for us? - Because they all exist, but we are only aware of this one with us in it.

I thought this one didn't exist, but was mere potential.

In space-time the billiard ball always exists in a state of not having been placed upon the table yet, sitting on the table, being struck for the first time...Because space-time is set (predetermined) and the flow of time is an illusion. The billiard ball possesses properties for everything that ever happens to it in space-time.

The billiard ball is said to be a worldline in this example, not moving through spacetime, but existing in a path within it.

Now if we take the billiard ball out of space-time, and place it in a dimension of time that is not set (predetermined), then we can either say that it has no future, or that it possesses an infinite number of potential futures

Or it could still have exactly one determined future. All you've done is changed time from block to flowing, and that doesn't have an effect on determinism. Determinism can be true or false in both cases.

You may think that the velocity of the strike also comes into play here, but it doesn't, only the location of the initial strike is relevant. The velocity of the strike only becomes relevant in subsequently.

I do think that in the case of your singularity, since a singularity (or a sole-existant sphere for that matter) has no distinct locations, and thus can only be struck one way if the magnitude of the force is not a factor. There is only strike or not-strike. There is no strike differently, at least not the way I see it. OK, the billiard ball has distinct sides since it is perhaps on a table or something. It travels in a direction depending on the strike angle. Maybe the analogy is not applicable to the singularity.

Well, the video is using the word 'dimension' like some fantasy place where you might go, like Narnia. They use it interchangeably with 'universe' so it sort of means the same thing to them, but that usage of the word doesn't even gather a mention in dictionary.com.
The video gets off on the wrong foot by answering a question about the 'multiverse theory', of which there are several . They don't say what kind (leaving you guessing). Most are a single universe with multiple non-interacting worlds.
Max Tegmark enumerated them all:
Type 1 is distant places, a relational concept. A star 50 billion light years (BLY) distant is completely nonexistent to Earth, and we are nonexistent to them, but if the universe is 'played' like your DVD, their universe probably gets played as well despite the lack of an obvious inertial reference frame in which we both exist. These are thus technically 'worlds', not whole different universes. 'Multiverse theory' is probably not a reference to this one.
Type 2 is different inflation bubbles in quantum eternal inflation theory. The view is as close to a scientific theory as you are going to get since it explains empirical evidence. Under the theory, there are different bubbles of spacetime, with different numbers of space and time dimensions, and different settings for the various cosmological constants. The existence of one of the worlds (ours for instance) by no means has any effect on any of the others. There just isn't a paradox here.
Type 3 is MWI interpretation of quantum mechanics, which is definitely one universe with multiple semi-interacting worlds. You speak of it below. A split is not a creation event of a new universe since both worlds still interact with common elements. Like Type 1, the laws of physics are identical in this world and another branch, so there is no paradox about the existence of one world having some new rule that negates all the others.
Type 4 is true separate universes, completely separate structures not based on any common thing (like quantum mechanics which binds all the types above). All rules are off with type 4, so perhaps this is the multiverse theory spoken of, except it isn't a theory at all, just a metaphysical musing on our part.

If there is a common element binding them all, like a juke-box waiting to 'play' one of the DVDs, then it really is one juke-box universe with a bunch of objects, not a bunch of separate universes. If they're joined like that by the juke box, then sure, there might be a law that allows the playing of one DVD to destroy all the others. I still fail to see that as paradoxical.

I am not saying that the branching multiverse suggested by quantum mechanics isn't the case, it may well be. If I flip a coin, for me the result may be heads, but for an alternate me, created by that event, the result may be tails. However; for me, here, this me, the me that got heads, I will always get heads not matter how many times the DVD gets played.
The other me, the me that got tails, the me that came into existence when I got heads, will also always get tails, no matter how many times this DVD is played. — StuartL

Pretty much how it goes, except 'you' are the one who gets heads, and yes, the one who gets heads gets heads every time the DVD is played. That's a tautology.

And that may all prove to be BS, it makes no difference to my theory.

Not exactly clear as to what is your theory. I mean, you say the flow of time is an illusion, but then you say that the universe is like a DVD being played, which would be a flow of time, would it not?
And then you say the properties of the surface of a sphere are infinite, which seems completely unrelated to any of the other points in the list in which it appears.
So mostly I am attempting to figure out the paradox, which would benefit from a bit of formal argument where supposing X leads to a conclusion of not-X, which demonstrates at least inconsistency, or a paradox if also not-X leads to a conclusion of X.

No rule says that any of it must be the case, it just may be the case. It's a bizarre concept. What I am saying is that it's possible that the singularity that supposedly "Banged" never actually "Banged". For us to exist it never had to, and that if it did "Bang" that it would cease to exist.

You're saying that our own big bang made itself nonexistent? Or just asking us to suppose this? If the latter, for what purpose? If the answer is nonsensical, perhaps it is an invalid thing to suppose.

What I am saying is that our existence, and the place in which we perceive ourselves to exist, are in actuality merely potential futures (time singularities), for the singularity that never actually "Banged". DVDs in a DVD player with nobody about to press play.

That's not a paradox unless you insist that the DVD must be played for there to be a DVD. Harry Potter on the DVD does not change a single bit by the playing of the DVD. Harry is aware of how the DVD begins, but not that is is playing, or was ever played. I find that most logical. The playing of the DVD on the other hand is not, since it implies something outside playing it, and then is their DVD being played? That would be infinite regress. So why posit that it needs playing?

Glad you are still around after the post got no hits for an entire month. I never say it until MiloL bumped it.

Different 'potential universes/ time dimensions' would have their own physical laws in the same way that different 'actual universes/time dimensions' would. Each 'potential universes/time dimension' could vary very little from another, or could vary vastly from each other. — StuartL

This bit gives me a clue as to you usage of the term 'dimension'. You seem to picture them as different universes with different rules, sort of what you get with eternal inflation theory with each 'spacetime' being a bubble in the inflation material.

What I'm referring to in the billiard ball example is, that in the Space-Time dimension, the fact that the billiard ball is on the table at all, and any interactions that I may have with the billiard ball, are already determined. The instant the “Big Bang” occurred and our universe's time dimension came into existence, all of Space-Time was determined. The “where/when” of everything within the Universe was determined instantaneously. So the billiard ball never exists in a state of “Maybe I'll be interacted with” instead, it always exists in a state of “I will be interacted with, in this manner, at this time”.

OK, this seems to be a single-world hard determinism view. Quantum Mechanics suggests this is not the case (all events are probabilistic, not predictable at all), but there is no proof one way or another. The view is not invalid.

There is many-world determinism (everything is completely determined, but all possible outcomes are real). In this sense, all of Spacetime is determined at the big bang, but the billiard ball is still in a state of 'Maybe I'll be interacted with' since there are multiple potential futures, only some of which involve a specific interaction.

Then there is the non-deterministic view where events are truly random, or where there is interference from outside the universe, violations of what could otherwise be deterministic physics. As a relativist, I suppose I'm classified under the single-world random category as to what is real.

My views on determinism vs. indeterminism are, that although Space-Time is set (determined), it is set by the choices that you made/make/will make. Ergo, you cannot change your future, but you wouldn't anyway, because your future was set by your choices, and you are you, so you would never have made different choices. An alternate you in an alternate universe may have made alternate choices, but you here in this universe made the choices that you made, and will make the choices that you will make. The moment of your conception and the moment of your death still/already exist. The flow of time is an illusion.

I agree with all of this. To 'change' the future is an incoherent concept. Change means to alter state from some prior state to some different later state, say a candle changing from tall to short as it burns. The future is not something that is one way until you 'change' it to something else. It was never the first way then, so there is no difference that is the 'change'. This is an arguable point, since the first state could be expressed in a 'would have been had I not ...' sort of manner. My choice 'changed' it from this abstract would-otherwise-have-been state. But that state is completely abstract and nowhere real.

For me the Space-Time dimension is analogous to a DVD. When the DVD is played, you make an appearance on screen at some point, but you are already/still on the DVD whether you are on screen at the time or not, or whether the DVD is being played or not.

A completely determined DVD need not be played for there to be subjective reality to the inhabitants of the DVD. Harry Potter hates Snape regardless of the DVD being played. The playing only serves a purpose to whoever initiates the playing of it, an outside entity that wants to observe the story. That observer is completely undetectable to the inhabitants of the universe/DVD, so Harry cannot detect when his DVD is being played.

What the paradox is saying is, that if every possible time dimension exists, and it is possible for a time dimension to exist that cannot exist in a multiverse of time dimensions, then such a time dimension must also exist. Ergo, a paradox, both a multiverse of time dimensions, and a time dimension that cannot exist in a multiverse of time dimensions, cannot exist at the same time.

Here you really lose me again, mostly because I cannot figure out what this set of 'time dimensions' is. Perhaps I could understand the paradox if these terms were spelled out a little more clearly.

What I am saying is “Yes they can, and here's how...”. (Apparently physicists hold symposiums and seminars to discuss this paradox and cannot figure out how to resolve it. But anyway...I digress.)

Not a digression at all. If physicists actually do this (it is a philosophical topic, not a physics one), then there would be a link somewhere describing the issue and some of the sides taken.

If we return to my analogy of Space-Time being like a DVD, what I am saying is that 'Imaginary Time' is like a massive DVD player, containing every possible DVD. However; none of the DVDs are actually playing, and if someone did come along, select a DVD and start playing it, all of the other DVDs and the DVD player itself, would cease to exist.

Why? What rule says this must be the case?

If I have a billiard ball sitting on a billiard table, the ball is said to contain potential energy, that can be converted to kinetic energy if I strike it with another ball. Let us assume that tomorrow I am going to do just that. Because the ‘flow of time’ is an illusion, and space-time is set, the ball already contains properties for any and all interactions that I'm going to have with it tomorrow (its future is set).

Now, if I have another ball, on another table, in another dimension where the 'flow of time' is infinite, and time is not set: Does that ball have no future, or an infinite number of 'possible futures'? — StuartL

The first paragraph seems to describe a sort of eternal 4D block universe, vs. the second case which has a 3D spatial state that is 'the present'. OK, either view can be translated to the other. Both can be deterministic or not, so it is possible to have a future that already is, yet has multiple possible futures. On the other hand, presentism doesn't imply a set of multiple possible futures. The ball may not yet have been struck, but it is perhaps inevitable anyway.
So you seem to be exploring determinism/non-determinism more than block/present views of spacetime.

Secondly, 'potential energy' usually refers to something that is capable of accelerating due to gravity, such as a ball atop a hill. The billiard ball does not contain the energy that makes it moves tomorrow, but rather acquires that energy from the ball that strikes it. Just a terminology thing that doesn't seem to affect your point.

To me, although it may sound a little bizarre, the ball now contains an infinite number of 'potential future properties', one for every possible interaction in the dimension. Because, isn't saying that the ball no longer possesses 'potential future properties', that could be converted into actual futures, akin to saying that the ball no longer possesses potential energy, that could be converted into kinetic energy?

You are going to have to define 'dimension' here because you are using it in a bizarre way. To me, dimension is the mathematical one: A square has two dimensions, a cube has three, and spacetime has four. You really rely on this word below, and I have no idea what you're asking.

'If all possible dimensions exist, and a dimension for which a multiverse is impossible, is a possibility, shouldn't at least one such dimension exist?'

???? Like maybe longitude exists but not latitude? I have no idea how 'dimension' is being used in this context. It is in quotes, so perhaps it is the definition being defined by somebody else.

Perhaps you mean 'world', but even that doesn't fit this context. You seem to use the word to mean that different rules apply in them. Can two universes exist if one of them precludes the existence of the other? Something like that? I suppose that they wouldn't be separate universes if one is dictating the rules about the others.
Yes, a single-world hard deterministic universe is possible, and it precludes other worlds in that universe. But a non-deterministic universe is also possible, and it might have multiple worlds. I see no paradox with that. Not sure if this is what is being asked.

It says right in the OP, " Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details." That is the first line of the problem. — Jeremiah

Sorry. Don't remember reading that. I brought up the issue before.

She is never told it is Monday, there is no relevant self-locating information, and she knew there were only three possible awake periods before the experiment. Everything we know is everything she knows before the experiment therefore 1/3 is a prior. We don't have any privy information here. — Jeremiah

Oh good. I thought the OP didn't make this clear. The halvers would have it right if she went into this thing blind.

Michael, I see you're still going on about this. Spent a bit skimming the posts since I dropped off.

I’ve showed the reasoning multiple times. It’s the Kolmogorov definition of conditional probability. — Michael

You're doing it wrong. You are mixing probabilities from different times, different points of view, or from positions with different information.

Choose a point and stick with it. P(Heads) of 50% is true only before the coin is tossed but you seem to assign that probability to Beauty's POV when in a waking period. That is question begging (as I pointed out early in the thread) since that's the answer we're looking for, not the premise. At that time, it is not 50% from her POV nor the administrators, where it is 0% or 100%.

P(Monday|Heads) means "the probability that it's Monday given the fact that it's heads" — Michael

Oh crap. OK then. I really don't know how to read this stuff then. '|' means 'or' in my world, but they have that intersection symbol to mean that here. Union for 'and'.

Sorry again, but the posts are coming faster than I can actually absorb that web link that seems to assume (reasonably) that you already know the rudiments of the notation. For instance the 'unconditional probability' hyperlinks to a page that makes no mention of the term. Not helpful.

So what's the condition? P(Heads|Awake)? Well, let's apply the Kolmogorov definition again:

P(Heads|Awake)=P(Heads∩Awake)/P(Awake)
P(Heads|Awake)=0.5/1=0.5 — Michael

As you see, the quote is getting altered. Sorry.
I think you are mixing the probabilities that something will occur (or is the case, but unknown) with probabilities that something known has occurred. You seem to assign 1 to P(Awake) which is not the probability that you will be away, but rather the probability after the information about being awake has been completely (not just partially) conveyed.

Not sure if a computation of P(awake) is going to yield what we want. It seems undefined. Of course we will be awake at some point. She's not being asked if she's awake, but being awake is information nevertheless.

I don't see how you are applying the additional information of 'it isn't Tuesday/heads' into your computation.
— noAxioms

That's the P(Monday|Heads) = 1. — Michael

Excuse me if I am new to the notation. I read this as the probability of it being at least one of Monday or Heads is 1, but since it might be Tuesday/Tails, this is wrong. I would think the probability of Monday or Tails is certain.
Maybe I just don't know how to read the notation.

But it doesn't follow from that that each of the other three outcomes are equally likely. — Michael

Respond then to my post about her getting to wake up on Tuesday.Heads as well. It spells that out.

All four of those things were equal probability. If odds were 50 Monday heads, 25 each Monday tails and Tuesday Tails, then there would be 75% chance that it is Monday, despite the day also being the same odds as the coin toss. I say it is 66% Monday because 2 of the 3 remaining options (A and B) are Mondays.

I've done so multiple times:

Kolmogorov definition:

<sorry, the formatting rendered the quote unreadable> — Michael

OK, I didn't know that's what that was. I'm actually not much up on the notation of it all, so I have a helluva time following it.
In the linked page, B is listed as unconditional probability. So yes, unconditional odds of heads is 50%, but Beauty is not working from unconditional, and I don't see how you are applying the additional information of 'it isn't Tuesday/heads' into your computation.

Why? If the flip had a 50% chance of being heads and if heads guarantees Monday then there's a 50% chance that today is heads and Monday. You seem to just be asserting these probabilities without adequately explaining how you got there. I get to my probabilities by applying an axiom of probability. — Michael

Perhaps quote the axiom in question. I suspect it applies to something not known, such as a coin toss that has not yet been made, or which has been put under a cup without observation. Beauty has been given information about the toss, and that cannot leave the odds unaltered. She has been informed that the combination of Tuesday and Heads is not the case. That information could not have been conveyed if the coin toss was still under the cup. If she had full information ("it was tails"), then the odds would change to 100% tails, axiom of probability not withstanding.

There's a 50% chance that both B and D will happen, but there's a 25% chance that today is B. — Michael

'will happen' implies the coin has not yet been tossed. It has, and today has happened, which yields information that changes the credence. That changes the odds of B and D to 66%, 33% each.

Suppose we change the game slightly. Beauty wakes up each day, regardless of the coin toss. But if it Tuesday and Heads, we tell her it is Tuesday and Heads. Otherwise we tell here nothing. Each time we do the brain wipe and go back to sleep. Beauty knows ahead of time we are going to do this (the whole 33% odds business depends on Beauty knowing the scenario).

I say nothing has changed. If it is Tuesday and Heads, she is informed of that situation, and is therefore not allowed to place a bet on the known coin toss. All of A B C D occur with equal probability, and any given waking has 25% odds of being any of the 4 (A,B,C,D) or 50% odds of eventual occurrence. But on C, she is told it is C. On the other three, she is not informed which waking it is. All she knows is that it is not C. 33% odds of being any of the other ones.

I ask now what the odds of heads is when she's informed that it is not scenario C this time? Because this is exactly the information she has been given.

There are three awakenings (A, B, D), each with equal 50% probability of eventual occurrence. In that sense, each awakening has an objective 50% probability of happening, depending on the coin toss.
But Beauty is apparently being asked not if this occurrence will happen (that is obviously certain during any particular waking), but rather which of A, B, D it is. Each has a 1/3 chance if their odds are equal, and they are equal since each equally has a 50% chance of eventual occurrence. B and D do not have 25% (less than A) chance of occurrence, as you seem posit. All three have equal probability, but only A is heads.

She gets one bet

And if you tell me that I'll get £1 for guessing correctly, and that you'll let me guess twice if it's tails, — Michael

This is a different scenario. When do I find out that I get a second guess? You would have to either tell me before the first guess, or after it. If after, odds are 50% on the first bet and 100% on the 2nd. If before, then 100% on both.
Beauty only gets one guess. If she didn't have the memory-wipe, then the situation would be as you describe it here. She would only have the information she needs on Tuesday, when the odds are 100% tails.

If I offer you one free lottery ticket if you correctly guess heads and two free lottery tickets if you correctly guess tails then tails is the better bet even though equally likely.
Tails isn't the better bet because it's more likely but because it has a better payout. — Michael

This is not the situation faced by Beauty. She is offered one bet that wins or loses one ticket. There is no double payout during any of here wakings.

It's not. It's about her credence. — Michael

Yes, it is about credence. Beauty has information about the coin toss, and that alters the credence from the 50/50 credence that exists to nobody in the scenario.

If you tell me that you flipped a coin ten minutes ago I'm going to say that there's a 50% chance that it landed heads. — Michael

Yes, because I've been given no more information, so the odds remain 50%.

That doesn't mean it's more likely to be tails. It just means that tails is the better bet. — Michael

You say it is 50% odds, but tails is the better bet. This seems contradictory to me.
The whole point of my answer was which was the better bet. If the odds were 2:1 instead of even (33% heads), then Beauty would make or lose no money on average by betting.

The odds of the flip are 50% from nobody's point of view. They seems to be 33% (the one point where neither is the better bet) from Beauty's POV, and they are 100% from everybody else's POV since they know the outcome of the flip during any of the wakings. It is 50% only from the external POV (not Beauty) only before the toss, which is not during any of the wakings.

It doesn't happen twice to her. It only happens once to her, given that it was only tossed once. She just wakes up to it twice. — Michael

And in what way is that not happening twice to her? She gets to bet twice in that case, despite the fact that she is unaware of which times she's betting twice.

Sure, but it doesn't mean that tails is twice as likely to occur as heads, — Michael

It is twice as likely to occur to miss Beauty. Tails happens twice, and heads only once.

So, I ran 100,000 games and gave 1 point for successfully guessing heads and 0.5 points for successfully guessing tails (because you get two opportunities). It doesn't matter if you always select heads, always select tails, select tails 1/2 the time, or select tails 2/3 of the time. The average score is 0.5 in every case. — Michael

Ouch... Does miss Beauty know that she's getting fewer points on some of the bets. In that case the outcome is certain and she can win every bet. If not, she's not the one doing the gambling.

This whole betting thing demonstrates the correct answer. It only needs to be done twice, not 100000 times, because there are only two unique cases occurring in equal probability.
I wake up and am expecting to bet a coin with even odds. I bet tails and win 2 coins (one each day) if it is tails, and lose one coin on Monday if it is heads. Sounds like a winning bet to me.

If the stakes depend on the coin toss, then I bet tails if the stakes are 0.5 coins per bet, and heads if the stakes are 1 coin in the bet. I win every time in that case.

I noticed a discrepancy.
The description of the problem does not make it clear that sleeping beauty knows the procedure. If she doesn't, then she has no knowledge of anything other than their asking about what is a random coin toss. The odds would be 50% then.
The 33% comes from the sleeper knowing that there will not be a waking on Tuesday if the result is heads, but there will be a waking on the other three scenarios. In this case, new information is gained (it is not Tuesday/heads), and the odds are not 50/50

Another simple way to compute this, objective, but without knowledge:
There are four equal probabiltiy states (each equally has 50% chance of being visited eventually, all depending on the coin toss. Monday and Tuesday are eventual certainties):
A Monday Heads
B Monday Tails
C Tuesday Heads
D Tuesday Tails

The sleeper wakes up and knows not which of the four it is, but she has the additional knowledge (new information) that it is not C, so 33% chance of each of the other choices, and only one of those is heads. Odds are 33%

Slow reply, I know. Been away.

This one is begging a different answer. From Sleeper's perspective, this has not been established.
— noAxioms

She already knows that it was a fair coin toss and that a fair coin toss has a 50% chance of landing heads. Nothing can change that. — Michael

Information about the toss can very much change that. If she is able to actually see the result of the toss, the odds become a certainty one way or another, not 50/50. So information does change the odds, and she has information beyond the simple fact that a coin was tossed.

To assert 50/50 odds here is to use a perspective other than that of the sleeper.

I don't know what you mean by adding up to 33%.

P(Heads) = 50% — Michael

This one is begging a different answer. From Sleeper's perspective, this has not been established.

But one of those awakenings is twice as likely as each of the other two, which is why the halfer answer is correct. — Michael

Yes, the Monday awakening is twice as likely as the Tuesday one. It doubles the weight of that awakening. So it adds up to 33% since there is a 50/50 shot on the heavier awakening, and a 0% shot on the Tuesday one.

So we have:

H: A, S
T: A, A

So since there are three possible awakenings and only one is when the coin comes up heads, then won't that mean she has a 33% chance of it being heads? — Jeremiah

I hate to say there is no conundrum about this one. I've not read most of the posts past this point in the thread.
Jeremiah's answer here seems correct to me. The 50/50 answer is from one who has no additional data and knows only that there is a coin flip. Sleeper here has additional data: I am awake, which reduces the case to which of the three awakenings above is this one.

Before you are born is the period I am referring to. If I start to exist there is the question of how I start to exist as that person. — Andrew4Handel

Well as I said, I think I am the wrong person to give answers to questions about a view that I do not hold.

When does combustion start to exist as a specific flame? Strange way to word it...

You could not be anyone else now but you could have been someone else and been born in another body or era or gender. — Andrew4Handel

Yes, that's the second guy I'm referring to. I say it doesn't exist. There is nothing that could have been somebody else until being born in this specific body. Mind you (pun intended), I am not asserting this. I'm just saying that the question you pose goes away with my answer. If this dualistic view is one you prefer, fine. As I said, religion has a lot of answers to how you got to be the fourth of six siblings, or how you got to be human at all for that matter.

I am one of 6 children I am conscious of being the fourth child but why not the first or sixth?

My answer is that it would seem absurd for the 4th child to be conscious of being the second child. That's what monism says. I think a lot of people that claim to be monists actually don't understand it and cannot accept that simple answer. It sure took me a long time.

That's why my first post dove into that 'objective' rant. The whole thing is easier to grasp from the outside in. Don't propose reasons P why I experience such and such. Ask instead that if proposal P is true, what would be the experience of person X? Same question, but more in 3rd person, and it yields different answers given the different assumptions built into the different perspective.

Never said I wasn't a person. I said I'm not two of them.

There is no realistic way of taking the "I" out of any theory because that raises the question of who is talking and what they are talking about. — Andrew4Handel

Fine. I have never heard of anybody that was somebody else. Why am I me? Well, who else could I be?

More your wording:

inhabit this conscious location of having experiences of a reality and how this subjective location arises — Andrew4Handel

'Inhabit'. OK, I see the path you want. Never mind at all what I say then. Religion has better answers to this one than I do.

But concerning location, I've also never know a person to be conscious at a different location than where they were. Both questions seem absurd to me, because there is only me, not two things paired by an 'inhabit' relationship.

The question already has biases built in, so you're on your own answering it. I struggled for quite some time figuring out this one until I identified the bias and the source of it. I found myself to be an improbable thing to be since there are so many other things (a bird, a stick, the duration of a flame), but here I am a well-off member of the species at the top of the food chain during the 2nd gilded age, pretty much the perfect thing to be. Baffling until I removed that bias.
So instead, don't assume that there is an 'I' that got to be 'me', or got to be 'here', and the problem vanishes.

As Thomas Nagel says "Objectivity is a view from nowhere" — Andrew4Handel

Well, an objective viewpoint is a view from nowhere and thus not really a viewpoint, but an objective description need not be precluded, and it often yields answers that elude a subjective description. You're going to need it for answers like this one.

I cannot render a 'view' (a drawing say) of a car without choosing a perspective, but I can describe one in full detail without the necessity of selection of a perspective.

Well this is not my paradox, I didn't invent it. It is a well known paradox, and widely recognized as such. Also the mathematical proof is posted in the OP. — Jeremiah

So you don't think is a paradox, OK fine, I don't really care. — Jeremiah

Fair enough. The relevant definition of paradox that pops up says this:

a : a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true
b : a self-contradictory statement that at first seems true
c : an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises — Webster

The funny cone seems to fall under definition 'a' since it seems opposed to common sense to many people. So yes, it makes sense to 'resolve' such paradoxes by showing that the seeming contradiction is something that is actually the case. The mathematics (a computation of the area and volume) is linked in the OP, but not sure what part of that is a 'proof' of something.

'b' seems to be the opposite of 'a': something that seems true at first but false on closer inspection.

I guess my idea of a paradox falls under 'c', the most basic example being "This statement is false". Any truth value assigned to that seems to be incorrect. I've seen it resolved in law-of-form using an imaginary truth value (square root of false) just like imaginary numbers solve square root of -1. There is application for such logic in quantum computing.

Saying there is no mathematical basis for this just tells me you can't read the math, as it is posted right there for you to review.

I never contested the mathematics, which simply shows that the object indeed has infinite area but finite volume. I can think of more trivial objects that are finite in one way but infinite in another, and your cone did not strike me as a connundrum. But I retract my assertion that it is not a paradox. The definition above speaks.

You can say, well it is not in the real world — Jeremiah

Indeed, it is only a mathematical object. A real one could not be implemented, growing too thin to insert ice cream particles after a while.

Interestingly, a liter of physical paint contains insufficient paint to actually cover a square meter of surface. There is a finite quantity of fundamental particles making up the volume of paint, and no fundamental particle has ever been found that occupies actual volume. So the paint is all empty space with effectively dimensionless objects which are incapable of being arranged to cover a given area without gaps. Instead, paint atoms work by deflecting light and water and such using its EM properties, not by actually covering a surface without gaps. Point is that this particular mathematical object has little relevance to even a hypothetical physical object.

Any container or solid object that has an endless surface area, but a finite volume is paradoxical, abstractly or otherwise.

Volume is the amount of space it takes up, so if it has endless surface area it should have endless volume. — Jeremiah

This assertion is exactly that: just an assertion, and a false one at that. There is no mathematical basis for this. The paradox apparently comes from your assumption of this nonexistent law.

The horn both converges and diverges, so it fits your personal take on what is needed for a paradox. — Jeremiah

A paradox is usually of the form of "If A is true, then A can be shown to be false". Your original 25 25 50 60 thingy would have been paradoxical had the 60 entry read 0%. What you seem to be reaching for here is not a paradox, but rather a violation of the law of non-contradiction, that a thing cannot be both X and not-X at the same time in the same way. I don't see the violation due to the 'in the same way' part.

So you are suggesting a finite amount of paint that goes on forever.

that paints an infinite surface. 'goes on forever' is not what I said, and seems a sort of undefined wording.
The alternative is that there is some points along your surface that do not enclose volume and are thus not painted.

So in your suggestion the volume of the paint both converges and diverges?

No, the volume is finite. You said that. There is finite (convergent as you put it) volume of ice cream, which could be paint.

I think you just can't admit that you were wrong. — Jeremiah

That is not philosophy, not by a long shot and if that is the standard that passes on these forums, then I have to question if I belong here at all. — Jeremiah

Because my arumgent is sound, besides I hate it when everyone sits around agreeing with each other, it is incredibly unproductive. — Jeremiah

This popped up this morning:
https://www.gocomics.com/fminus/2018/05/23

Thought it expressed my takeaway on this topic.

This one is a bit trickier and as far as I know it has not been resolved. — Jeremiah

In what way is this in need of 'resolution'? You haven't stated a problem with this scenario.
Is there some law somewhere being broken, like infinite surfaces must enclose infinite space? There is obviously no such law, as demonstrated by this example.

So you are suggesting if it was filled with paint, you could use a finite amount of paint to paint an endless surface.

It seems to me, that you'd run out of paint, and even if you could stretch the paint infinitely thinner, that still does not resolve the paradox. As abstractly what you have is a cone with a converging volume and a diverging surface area. — Jeremiah

Clearly the paint would not run out, as it hasn't in your example. It covers the entire surface, and doesn't even need to be spread out to do so, since it has finite thickness (all the way to the center line) at any point being painted.

I see no paradox in need of resolution. The volume converges and something different (the area) does not. It is only paradoxical if the same thing both converges and diverges.

So what would constitute a proof that your first assertion is wrong, what kind of proof would we need to present, and what standards would be assessing that proof by to see if it held? — Pseudonym

A simple example to the contrary suffices in proving wrong an assertion that all A is B, or in this case, the only valid interpretation (A) is one of a sample space of 4 (B). Many of us have produced that alternate interpretation of a sample space of three (~B).
That leaves it up to Jerimiah to demonstrate that either this sample space is not 3 in number, or that one has no way of randomly selecting from this sample space. The latter would be the case if the answers were hidden, but the OP concerned an open multiple choice question, not drawing of hidden names from a bag.